Theorem rgen2a 1264

Description: Generalization rule for restricted quantification.

Hypothesis

Ref	Expression
rgen2a.1	|- ((x e. A /\ y e. B) -> ph)

Assertion

Ref	Expression
rgen2a	|- A.x e. A A.y e. B ph

Distinct variable group(s):   x,y   y,A

Proof of Theorem rgen2a

Step	Hyp	Ref	Expression
1		rgen2a.1	. . . 4 |- ((x e. A /\ y e. B) -> ph)
2	1	exp 291	. . 3 |- (x e. A -> (y e. B -> ph))
3	2	r19.21aiv 1259	. 2 |- (x e. A -> A.y e. B ph)
4	3	rgen 1247	1 |- A.x e. A A.y e. B ph

Syntax hints:   -> wi 2   /\ wa 196   e. wcel 1092  A.wral 1201

This theorem is referenced by:  f1stres 3096  ruclem13 4897  helch 5151  hsn0elch 5155  shscl 5282  shintcl 5294

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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